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Chapter 5 Visual computations

An application of this methodology in formal analysis is attempted here using Richard Meier’s

Smith House as a case study. All plans of the house are represented and decomposed in specific

ways as described in the previous chapter and the computation of all symmetry parts takes place

in entirely visual terms. A reassembly of the layered symmetries explains the structure of the

symmetry of the house and provides an illustration of the basic thesis of this research on the

foundation of a theory of emergence based on symmetry considerations.

5.1. Introduction

‘Were architecture to be a dream of pure structure, Eisenman is the one who, more

than any other in America, comes closest to achieving it; if, however, architecture is

a “system of systems”, if its expressions belong to different but interwoven areas of

language, then it is Meier who is able to grasp those relationships.’ (Tafuri 1976)

The 1967 Exhibition New York Five (NY5) on the early work of five New York city architects,

namely Peter Eisenman, Michael Graves, Charles Gwathmey, John Hejduk and Richard Meier,

and the subsequent book Five Architects published in 1972, have indelibly stamped the course of

the history of modern architecture of the late twentieth and early twenty-first century. The explicit

reference of NY5 to the work of Le Corbusier in the 1920s and 1930s and its ironic allegiance to

a pure form of architectural modernism made the exhibition pivotal for the evolution of

architecture thought and language in the subsequent years and produced a critical benchmark

against which other architecture theories of postmodernism, deconstructivism, neo-modernism

and others have referred, critiqued or subverted (Tafuri 1976; Jencks 1990; Major 2001). Among

this early work of NY5 the Meier's buildings were closer from all on the modernist aesthetic of

the Corbusian form and in fact even the later buildings that Meier produced since then have all

remained truest to this aesthetic.

The work here traces the history and logic of the evolution of Meier’s early language

and its direct relationships to spatial and formal investigations of early twentieth-century

modernism as well as its direct reciprocal relationships with the rest of the NY5 languages. The

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departure for this inquiry of such centrifugal relationships between rules and products and

between notation and performance, for the purposes of this work is Richard Meier’s Smith House,

an early pivotal work and acknowledged forerunner and embodiment of the full repertory of

Meier formal strategies and language (Colomina 2001).

The formal theory for the analysis of the language of the house is based on the model

developed in the previous chapter. The specific methodology for performing this analysis relies

on partial order lattice representations for the decomposition of a design (Park 2000; Economou

2001). Here this methodology is extended to describe complex spatial configurations

characterized by architectural concepts such as layering transparency and collage (Slutzky 1989;

Hildner 1997). More specifically, the analysis here uses all three levels of representations

postulated by the model for the description of the house, each one specifically designed to bring

forward different aspects of its spatial composition. The partial order lattice pictorially presents

the symmetry structure of a rectangle-based spatial configuration. The number and qualities of the

symmetry subgroups found in Meier’s architectural composition provides the maximum number

of layers. Analytically, these layers of the architectural design are used to reveal parts and sub-

symmetries that are used strategically or the scaffolding of the design. Synthetically, group theory

suggests operations and spatial transformations that may have been in compositional and thematic

development of the design. A partial ordering of sub-symmetries and a classification by lattice

diagrams of sub-symmetries exposes the underlying structure of the complex designs.

A major motivation of this work is that there is a correspondence between the evolution

of architectural languages and the formalisms that can be used to describe, interpret and evaluate

them. Classical modern buildings can be and have already been successfully described by group

theoretical techniques. In the same way, Richard Meier’s work constitutes a hyper-refinement of

the modernist imagery that has been inspired not by machines but by other architecture that was

inspired by machines and especially Le Corbusier (Goldberger 1999). Thus, the group formalism

can describe Meier’s architecture as a hyper-refined construction that relies on specific

representations and mappings that foreground internal complex relationships of the structure

itself, i.e. the symmetry subgroups and super-groups of any given spatial configuration. Here, all

plans of the house are analyzed in terms of corresponding group structures and all are represented

in partial order lattices using axonometric orthographic projections to illustrate notions of

complexity, ambiguity and emergence and the ways they all inform design.

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5.2. White geometries

‘I would suggest that what distinguishes the Whites from Le Corbusier lies precisely

in their elevation of form from the condition of design to that of epistemology. What

was at stake was the claim that form was a type of knowledge; indeed, an essential

type of knowledge.’ Deamer (2001)

To understand the formulation of the formalist epistemology architecture takes under the New

York Five, also known as the Whites, one needs to survey their legacy on one hand, and on the

other, to understand the fundamental emphasis of the homogenous plane plays on their work on

the surface. To understand this formulation, Deamer’s (2001) account tells us that ‘The true

legacy of the Whites is not their formal vocabulary... but the fact that these (formal) operations

have a systemic intellectual import at all.” The linking of form to knowledge is due to Hildebrand

(1893), and later Arnheim (1954), but what is new is its import to late modernism.

The initial core of shared intellectual aspiration is formed in 1954-56 by the association

of Colin Rowe, Robert Slutzky, and John Hejduk as “Texas Rangers” at Austin, and later Peter

Eisenman with Colin Rowe at Cambridge in England (Caragonne 1995). While Rowe brings with

him the legacy of Rudolf Wittkower who sees Renaissance as a scientific program, Slutzky as a

painter brings the Arnheimian idea of depth on the picture frame, frontality as the dominant visual

ordering system, and strong separations between foreground figure and background field. Graves,

Hejduk, and Meier began their architectural search with their concern for organizing the visual

world as in painting in a spatially complex manner. What the Whites have in common is to inject

the most subtle commentaries on spatial layering, frontality, rotation, skews and ambiguity to the

debate of the International Style. Some of their representative works are shown in Figure 5-1.

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Figure 5-1: Representative work of NY5: a) P. Eisenman, House II; b) M. Graves, Hanselman House;

c) C. Gwathmey, Cohn Residence; d) R. Meier, Smith House; e) J. Hejduk, House 10

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Eisenman in House II in 1969 develops systematic analytic diagrams combining basic

formal moves to sophisticated results (Eisenman 2004). Starting with a square volume marked by

a matrix of sixteen square columns, its underlying structure is expanded by transformations based

on sequential layering of solids and voids. Any element or relationship between elements has two

notations, marks, or weightings of relative equivalence. He furthermore creates a sense of

ambiguity between figure-ground, solid-void, window-wall though the definition of the elements

as well as though their placement, size, and number Figure 5-1a).

Graves in Hanselmann house in 1967 uses abstraction before migrating through collage

to the poetics of neo-classical motifs (Graves 1982). The house is understood frontally by the

layering of three principal facades. The first consists of a pipe rail frame and the front plane of a

studio house. It acts as a gate, receiving the stair between the ground and the entrance level. The

second primary façade, located at the center of the composition, contains the point of penetration

into the main volume of the house. The third façade which is the densest is the real wall of the

house’s composition and the surrounding landscape. An outer terrace relates to the diagonal of

the stream and implies a larger compositional frame (Figure 5-1b).

Gwathmey’s abstract formal vocabulary in Cohn residence in 1967 is devoted to the

interaction or interpenetration of contrasting platonic volumes and pure shapes (Gwathmey and

Siegel 1984). Form relates to the line of interpretation between abstraction and representation.

The work appears to rest in the Cubist frame of reference. Clearly, architecture is not skin-deed.

To detach the bones from the skin is the beginning of formal irresolutions which deny the basic

principles of the composite overlay of plan, section, façade that produce the building (Figure

5-1c).

Meier in the Smith house in 1967 uses geometry as a magnification of architectural

functions with objects which display their function in absolute clarity (Meier 1984). The house

has a layered structure, in which the relationships between volumetric order and transparency,

and the analysis of possible geometric articulations suggest certain analogies to the structural

purity of Eisenman and even to some ambiguous metaphors of Graves. Meier (1984) offers an

architecture that presents itself as ‘a system of systems’ (Figure 5-1d).

Hejduk in House 10 in 1966 studies the formal propositions of the avant-garde to draw

imaginary projects (Hejduk and Henderson 1988). ‘To fabricate a house is to make an illusion’.

In House 10, basic geometric forms (circle, square, diamond) are cut into quarters and are

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separated and grouped at the ends of a long path. By doing so, Hejduk performs two

complementary tasks: he chooses absolutely trivial forms, and uses the technique of volume

deformation and that of sectioned volumes according to elementary rules (Figure 5-1e).

In all, clear compositional strategies to eliminate the vicissitudes of subjective seeing

mark the work of the Whites: frontal-rotational, solid/void, grid/dissolution of the grid,

virtual/actual solids and voids; whole and partial Platonic figures, regulating lines, datum, and

golden proportions. This process of abstraction goes hand-in-hand with a concept of

defamiliarization through the elements of the frame or the grid creating a Cartesian field. Thus,

the function of the plane as a method of stratifying space implies a kind of wall whose primary

function is to modulate space. The Whites share an enduring interest in Le Corbusier and in turn

provoke the creation of an oppositional group, the Grays, who promote a less abstract

architecture. Colin Rowe and Vincent Scully are supposed to be their respective backers. The

difference between the Grays and the Whites is the supposed privileging of perception by the

former and conception by the latter, while in fact ‘the Whites have usurped perception to their

own ends, making it a conceptual tool’ (Deamer). But the operations that link them to Le

Corbusier start with the grid dominated by field and figure that provide the framework for

operations of transformations. What distinguishes the Whites from Le Corbusier, is namely their

elevation of form from the condition of design to that of epistemology. As Deamer (2001) puts it,

‘Le Corbusier never identified form in and of itself as the ends of architecture... One would never

find in Le Corbusier, or any of the original modern architects, arguing, as Hejduk does, about the

essential merit of the diamond over the square, or the necessity of revealing deep form in the

environment, as Eisenman does... For the Whites, there is the relationship between percept(ion)

and concept(ion) – if architecture is a form of thought, how does visual perception interface with

that mental construct?’

The breakthrough toward this epistemological surge on formalism yields the reliance

on forms of thought exterior to architecture, be they philosophical or scientific, half-materializing

Lionel March’s call for his adoption of scientific models for architecture. For Deamer, one of this

phenomenological strain is the rise of Daniel Libeskind, Hejduk’s former student, who combines

Hejduk’s poetry with Eisenman’s conceptual logic. Greg Lynn, Eisenman’s former student,

developed his biological methods within the same framework. Needless to say that Cache’s

epistemology or Berkel’s conceptual techniques are the offsprings of the theoretical territory

prepared by the Whites (Deamer).

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The formal framework for understanding the underlying connection between the

Whites and contemporary architects has also been facilitated by the rise of CAD systems in

design that help link thought and images, perception and conception. It seems that the

organizational scaffolding corresponding to the way the Whites link the forms of architecture to

the layering is so algorithmic that their generative reliance on numerical predictability attracts

contemporary designers to inherit the Whites’ design methods via computer versatility (Deamer).

This move is also facilitated by the current trend to emphasize work on the surface as

the primary conceptual device – it is the surface Libeskind writes on, the surface Lynn folds, the

surface the computer turns into topography for Cache. The planes deployed by the Whites are the

phenomenal datum onto which three-dimensional spaces collapse. The planes deployed by the

digital architects mostly are not. One can read the most recent surface work as an emancipation of

painterly two-dimensional surface from the no longer dominant grid and volume. That kind of

work is not a-spatial per se, but the complex organizations of physical matter yield complex

spatial interiors. Although the spaces are never conceived of positively, the common trait

remains: complexity and ambiguity. The Finnish architects conclusively make a comment on the

added meaning of the simplified box that gives the spectator a new variation and a wider

possibility of viewing the third dimension more completely. The combined effect of layers of

different elements and materials creates a new kind of homogeneity. What Reima Pietila explains:

“These model house developments of the sixties and seventies have quite another feeling for form

compared with Le Corbusier’s twenties: that of cold computer intelligence… The Futurist present

of that great mystic Le Corbusier has come to nothing and has been replaced by the new

metamorphoses of continued space and the infinite permutations of the sixties.” (Stenros 1987)

5.3. The Smith House

‘Meier’s hyper-refinement of the modernist imagery has been inspired not by

machines but by other architecture that was inspired by machines… honoring his

“fathers” and casting them off at the same time.’ Goldberger (1999)

It is clear that the complexity suggested in the reading of the corpus of the NY5 would make any

of the buildings belonging in the set an ideal candidate for a case study for the analytical method

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developed here. Still, it is argued here that among all possible candidates, the Smith House stands

out as the right candidate. The house has a long legacy: Frampton and Johnson (1975) nominated

the Smith House as a classic and selected the young Meier as the one architect out of five who

knows history the most and learns from it; Rykwert (1999) has asserted that the house is a classic

case of one that has been designed upon a formal vocabulary whose elements are all abstracted

from the repertory of early modernism and juxtaposed back as a collage; Jencks (1990) has

asserted that Meier uses a mixture of traditional forms of modernist architecture and where the

system is incomplete, new elements are added. And still many other key discourses have been

suggested to include the themes of compositional grid and patterned frames (Goldberger 1999),

the discipline of the Dom-ino and Citrohan structures (Kupper 1977), Mies’ aesthetic of rhythmic

linear elements (Hildner 1997), and several others. It is clear that Meier's language, iconography,

and elemental categories force comparison and differentiation with the work of the other

members. Meier’s long standing personal affiliations with artists and his torn-paper collages are

his technical link to the painterly means of space-making by the use of color, surface, line, and

contour.

For the purposes of this research, and in addition to what has been mentioned so forth

about the house, a key aspect of the house is that it embodies the quintessential aspects of the

abstract modernist vocabulary in that it exemplifies the organization of space through the abstract

instruments of plan and section. The Smith house is the first of Richard Meier’s white buildings,

which the architect characterizes as: "the precise manipulation of geometry in light that translates

into power of architecture to become art" (Meier, 1996). Most importantly, it is the project that

exemplifies the most the wall not as an element of construction but as an abstraction, as a spatial

element, a homogeneous plane (Meier, 1988). The architect considers this house his first mature

building: “It was there ... that I was first able to develop and test a number of issues that I had

been preoccupied with. In fact, those issues still preoccupy me: the making of space, the

distinction between interior and exterior space, the play of light and shadow, the different ways in

which a building exists in the natural or urban world; the separation of public and private space”

(Meier 2000).

5.3.1. A first encounter: Site Structuring

The Smith House is in Darren, Connecticut, and it is situated on a 1.5 acre site overlooking Long

Island Sound from the Connecticut coast. The house was built during 1965-67 on a site literally

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adjacent to the water and it was designed for a family with two children. The site plan of the

house is shown in Figure 5-2.

Figure 5-2: Site plan of the Smith House

The house is developed over three levels. The entrance area and master bedroom are on the

middle floor. The lower level is for dining, kitchen, laundry and domestic help. Both the living

and dining areas open directly to outdoor terraces. The top floor contains children’s bedrooms,

guest-room and library-play. The house is finally topped by an outdoor roof deck. The three plans

of the house are shown in Figure 5-3.

a. b. c.

Figure 5-3: Plans of the Smith House. a) Lower floor; b) Middle floor; c) Upper floor

The house itself appears to be a hyphenation of two canonical structures: the Citrohan house and

the Domino house (Corbusier and Jeanneret 1937). The Citrohan zone is a series of closed

cellular spaces and the Dom-ino zone is leveled as three platforms within a single volume

enclosed by a glass skin. Meier investigates a language of oppositions of a denied dialectic

between the total transparency of the panoramic façade and the solid compartment of the entrance

façade. The handling of the layer stratification of the building parallels the post-Cubist conception

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of spatial relationships. On this basis, the conception of the spatial arrangement of the house too

parallels the development of combinations and assemblages of lines, planes and volumes,

independent of what the given elements may represent. Two facades and sections of the Smith

house are shown in Figure 5-4.

a. b.

Figure 5-4: Orthographic views of the Smith house. a) Longitudinal section; b) Transversal section

5.3.2. Second encounter: Maximal lines

A typical framework for formal analysis is the identification of all possible regulating lines in

plans and facades and the examination of the characteristics of the emergent shapes and

configurations of the regulating lines. Typically such an analysis, especially for rectangular

geometries, proceeds along extensions of the walls to provide grids and shapes with special

characteristics, say squares, root–two rectangles, golden–section rectangles and so forth as well as

spatial relationships between them, say, center to center, center or edge, edge-to-edge, and so

forth. Several variations of these regulating frameworks are given in Figure 5-5.

Figure 5-5: Search for alternative partitions. a) Root-2 lines; b) Candidate centers

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All regulating lines provide several alternative partitionings of the house and they are

clearly based on the geometry of the rectangle. These rectangles come up in various sizes and

dimensions depending on what is subsumed under them. Still, it is clear that all those can be

grouped in three general classes of arrangements depending on the choice of center for the

selected rectangular configuration. Possible decompositions of the house include: a) a major

rectangular space with three secondary spaces attached to it in right angles relationships; a major

frontal rectangular space that is interspersed by the element of the staircase; or alternatively a

major rectangular space that subsumes all parts of the house and all spaces are thought of as

carved out from the major body of the house. Other readings are certainly possible. All these

decompositions suggest alternative solutions to the compositional problem of identifying, if there

is, a common appropriate center and axes of symmetry and disposition of the configurations. The

three cases explored above are given in Figure 5-6 as variations exploring the mimima and

maxima of bounding rectangles.

Figure 5-6: Three decompositions of the house in terms of a major rectangle a) Minimum; b) Medium; c) Maximum

The same regulating lines as extensions of walls and correspondences between them,

construct as well systems of lines and emergent grids that are used to interpret space according to

a pattern of oppositions: vertical/horizontal, top/bottom, orthogonal/diagonal, left/right. These

grids can be drafted on various directions foregrounding the x direction or the y or even diagonal

relationships between the cells. Three superimposed grid systems for the three levels of the house

are shown in Figure 5-7.

Figure 5-7: Superimposition of grids for the three levels of the house

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These grids show their ability to be used as organizations devices for the interpretation

of space when all walls are eliminated and the regulating lines are shown by themselves. These

scaffoldings compose then the underlying organizations structure of the house and foreground

arithmetical and commensurable relationships between the parallel lines. The three grid systems

extracted from the wall extensions for the three levels of the house are shown in Figure 5-8.

Figure 5-8: Rhythmic dispositions of grid

The same principle applied at the elevations and sections of the house suggests

somewhat similar and different interpretations. The most immediate finding is that the same root-

2 rectangles that can be found in the plan can also be found in the elevations and the sections.

That is not surprising given the abstract modernist vocabulary of the house and its

exemplification of the organization of space through the abstract instruments of plan and section.

What is more interesting is that the same trivalent condition (T-shape formation) of the

intersection of the lines in the plan exist in the façade too but now it is even more celebrated in

various ways constructing essentially grids in essential nested ways. The rules that can account

for such a T-shape intersection are straightforward as shown in Figure 5-9.

.

Figure 5-9: Rectangular divisions of the rectangle

Successive applications of such rules produce quintessential modernist arrangement

with nested grids populated by T-Shape intersections. A derivation of a typical nested T-shape

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grid is shown in Figure 5-10 and a series of T-shape intersections found in the façade and the

section of the Smith house are shown in Figure 5-11.

Figure 5-10: A derivation of a nested T-shape grid in the Smith house

Figure 5-11: Rhythmic dispositions of grid

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5.3.3. Third encounter: Planes and walls

‘I am interested in spatial elements, not elements of construction. I want the wall to

be a homogeneous plane.’ Meier in Davies (1988)

Here a third encounter with the description of the house is suggested and this time the wall

element is foregrounded. Meier himself has attested to his preference to spatial elements rather

than construction elements and especially his predilection for the wall to be a homogeneous

plane. A close examination of the instances of the wall in the house and their spatial relations

suggests compositional processes such as parameterization, dematerialization, deformation,

defragmentation and therefore point to the design of an additional overall framework for a critical

description and interpretation of the house. This suggestion here is based on a series of

experiments upon the representational elements of the house and their consistent typological

reduction in the planar unit of the wall. This is not as easy as it sounds because the geometric

rectangular prisms of the house resist their immediate interpretation: Often they appear as

massive blocks - space volumes, other as opaque walls with or without openings; and lastly as

emergent planar shapes that organize space. Different interpretations of these rectangular prisms

as walls of different kinds are shown in Figure 5-12.

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Figure 5-12: Different interpretations of rectangular prisms as space volumes, perforated walls, or compositional planar partis.

The basic topology of the wall can be seen as a planar rectangular element with cutout

parts to account for openings of all sorts, including doors, windows, and thresholds of several

conditions. The planar element can be modeled after a dimensionless gridiron pattern whose cells

may denote closed and open parts in the wall and the formal representation of the wall is thus

taken as a binary configuration based on an n × k grid. Among all possible gridiron systems the

3×3 was chosen here as the most generous for architectural purposes. The basic gridiron pattern

of the wall is shown in Figure 5-13.

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Figure 5-13: The cycles of permutations for the 3×3 cell

The task is to identify all possible non-equivalent combinations of closed and open cells that can

be found in this configuration and map these structures to the set of walls that Meier uses. In

other words, what is suggested here is that the specific decomposition of the three-dimensional

cuboid in Meier’s work can be seen as based on a decomposition of the two-dimensional square

into specific gridiron systems and that by de-fragmenting the modules of vertical planes to

determine the classes of openings in a planes, Meier’s palette can be easily put on display.

The method of counting of non-equivalent configurations based on a given permutation

group of vertices of a geometric shape has been given by Polya in his theorem of counting. The

description of the formalism and various applications in other domains has given elsewhere

((Polya and Tarjan 1983); Economou 1999; Din and Economou 2007). Here only the application

of the theorem on this specific context is given. The most critical part of the application is to

identify the model of the structure. The 3×3 grid is here taken as a 3×3 cellular structure whose

cells are represented by vertices. Polya’s theorem will then provide the answers about the binary

condition of these vertices. Figure 5-14 shows the remodeling of the nine lines into nine vertices.

Figure 5-14: The 3×3 grid represented as an array of 3×3 cells

The core of the theorem is that any shape can be represented as a function of the cycles

of permutations of vertices fr that here are induced under the symmetry group of the shape, here

the D4, the symmetry group of the square. Figure 5-15 shows all the eight cycles of permutations

induced by the eight symmetry operations of the square. It is worth noting that the vertical,

horizontal and diagonal symmetries induce the exact same permutation of the vertices and

different from the half-turn symmetries; this is quite different behavior from the permutations of

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the vertices of the square under the same operations where the half-turns induce the same

permutations as the vertical and horizontal but not the diagonal.

Figure 5-15: The cycles of permutations for the 3×3 cell

The sum of all the cycles of permutations and their products divided to the sum of

permutations of the symmetry group of the figure provides the cycle index of the figure, the

blueprint for the enumeration of all the possible subsets. Here, for a figure inventory x+y where x

and y represent the quantities that will be enumerated – the closeness and openness of the cells,

its expansion according to the theorem is given in (3).

f r = x r + y r (3)

If by substitution the figure inventory into the cycle index replaces the cycle fr with xr +

yr, and expands the cycle index in powers of x and y, the resulting coefficient of xryr is the number

of distinct ways of configuring the x cells and y cells with respect to the permutation group. The

equation can be solved in a straightforward way by taking advantage of the multinomial theorem

in (4).

sr

nsr

n yxsr

nyx!!

!)( (4)

In the specific case here for the 9-cell grid the computation of the equations (3) and (4)

for the figure inventory given in Figure 7 provides the complete set of solutions and that is a total

of 102 distinct configurations. These configurations are symmetric regarding the quantities x and

y. The 102 n-cell configurations for x + y 9 are shown in Figure 5-16.

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Figure 5-16: The 102 n-cell configurations of a 9 square-grid for x white and y black cells

The exciting part of this enumeration is that it provides the complete set of all possible

configurations of all binary systems embedded upon a given grid and therefore it provides a

systematic framework to explore all the possibilities implicit in the system. It is clear for

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example, that some of these configurations have been used in many different circumstances in the

design of the Smith house; these configurations consist of arrangements of black and white cells

that denote respectively open and closed spaces or some hybrid in-between spaces. All walls are

then understood as abstract geometrical cuboids exemplifying these abstract configurations as

shown in Figure 5-17.

Figure 5-17: Non-equivalent configurations of a 9 square-grid in Meier’s planar units

The parameterization of the bock produces variations in dimensions, density and edge

condition. Interesting cases emerge: A massive piece of wall or block can generate any of the

most unlike elements: chimney, closet, recess, threshold, staircase, and so on by subtractive

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operations. A solid opaque wall can be subject to operations of filtration, permeability or

translucence. A solid transparent wall may instantiate either a glazed curtain wall or a window

wall. Finally, a virtual wall as an abstract plane is then defined by its edge condition. A line on

the plan may mark the separation of inside–outside, but it can also signify the edge of the volume,

a change in material or level, or the presence of something above or beyond. A combination of a

solid - opaque wall and a transparent glazed wall may yield a translucent wall. A combination of

a solid wall and a space volume yields to a hybrid wall and so forth.

The exterior walls of the house can be nicely captured with these definitions. The

frontal wall of the house is a triplet of planar and volumetric elements imbued with materiality

and permeability. The glazing element incorporates open frames of wood| steel with an infill of

glass. The trabeated element plays the role of concentric shell which acts as another filter. In-

between is found an appended volume of space. The lateral window facade is layered same as the

frontal one. Hybrid units are layered in parallel. Here, all the enclosing walls are hybrid. In the

middle, there is the medial wall to create a vertical layer for the promenade architectural and to

structure deep and shallow space. The south frontal wall of the house is a triplet of planar and

volumetric elements imbued with materiality and permeability and generated by a block. The

glazing element incorporates open frames with infills of glass. In the middle, when there is the

double wall to create a vertical layer for the ‘promenade architectural’ and to structure deep and

shallow space.

The basic unit of the composition of the Smith house, the wall, with all its variations

can be seen a vocabulary that comprises a subset of a specific set of topological transformations

of rectangular prisms and correspondingly of the full vocabulary of the NY5 architecture.

5.3.4. Fourth encounter: Layers

According to Birkhoff (1932) the rectangular forms are best suited for use in composition. This is

an algorithm to construct the five subgroups of symmetry of the rectangle that will populate the

lattice. It will be made clear how they derive from one another, how some idea of weight or

precedence can be set, and how object-feature mapping is applied as a function after Ho (1982 d).

The search of the possible partitions of the rectangular frame of the building leads to multiple

choice configurations. An emphasis on the golden section rectangle is considered here as a

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canvas. This kind of rectangle seems the most appropriate among the other choices, and it

supports most of the axes of symmetry. The following steps report the result of this investigation.

Meier himself hints the starting block by providing the six partitions of the Citrohan

block ( 2 rectangle). The first function f1: V 1f V, maps of vertical reflection set into itself,

with a restriction upon the range of (V, f1) to the bottom half-plane so that,

||)(1 vvf such that if v 0, f1(v) = -v, and if v 0, f1(v) = v (5)

Figure 5-18: Vertical reflection V

The second function f2: V 2f H, maps horizontal reflection from the set V based

on the Citrohan part into the top half-plane sheltering the Domino part. Thus, another restriction

sets the range of (H, f2) to the top halfplane as a well-defined mapping. For any h in H there exists

v in V (onto function) such that,

hvf )(2 (6)

Figure 5-19: Horizontal reflection H

The third function f3: SH2f SV, maps rotation or half-turn or spin from the set H

belonging to Domino part into (top half-plane) into corresponding elements in the Citrohan part.

vh ssf )(3 is a rotation through angle ( = /2) such that cossinsincos (7)

Figure 5-20: Half-turn S

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The fourth function f4 maps the three horizontal layers corresponding to the layered

frontal, walls.

f4: D2= ),( HV 2f D2= ),( HV (8)

Figure 5-21: Dihedral D2

One subgroup is a placeholder for the elements that are unique in the house. These are

add-ons labeled C1. Call them -singularities, or -unlike elements. In the Smith house there are

five of them: chimney, interior staircase, exterior staircase, ramp, and cottage.

)(5 vf (9)

Figure 5-22: Singularities or Collage C

5.4. The Smith House: A formal description

The analysis proposed here proceeds along visual computations that are all based upon the

models of analysis presented so far and uses representations that capture some, but of course not

all, conventions characterizing a design. The key idea behind these computations is that they are

designed to decompose the house in sets of basic elements that are then recomposed to redescribe

the house and help interpret the basic assumptions about the system itself. The four aspects of

representation used are abstraction, projection, weights, and layering. All definitions have already

been given; here only a brief précis is given again: Abstraction captures the geometrical

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characteristics of the architecture object at different levels of detail. Projection denotes the

specific orthographical or oblique mapping of the model of the architecture object upon the plane

of depiction. Weight denotes the physical characteristics of the architectural object such as

opacity, translucency, or transparency and is captured by three types of lines used here: Solid,

thin, and dotted. Layering denotes the decomposition of the design in distinct parts. Other aspects

of architecture representation routinely used in architecture notation are omitted here.

5.4.1. Initial shape

The initial shape that starts the computation is the three-dimensional model of the Smith house

that is built upon existing plans, elevations and sections of the house. An axonometric view of the

three-dimensional model of the house is given in Figure 5-23.

Figure 5-23: Axonometric view of the Smith House

5.4.2. Rewind

Three levels of abstraction are considered here as generous enough to capture key stages in the

description and interpretation of the design. These levels are organized respectively as notational

languages that describe some but not all features of the architecture space.

The first level of notation, code-named here as ‘architectonic level’, is the level that

approximates in some way the original notation used for the language of the Smith House and for

the most part the rest of the NY5 designs. The notation privileges functional elements such as

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walls, slabs, columns, and beams, walled furniture, handrails, and openings of various kinds such

as windows, doors, stairwells, chimneys, and so forth. The architectonic representation of the

Smith House for all three floors is shown in Figure 5-24.

Figure 5-24: Architectonic level: a) Lower floor; b) Middle floor; c) Upper floor.

The second level of notation, code-named here as ‘spatial level’, privileges space

divisions and corresponding openings in these boundaries, and discards all other information.

This level essentially picks up planes that function as walls and slabs and so on, the connections

between them and their interface with context for ventilation, light and so forth. The spatial

representation for all three floors is shown in Figure 5-25.

Figure 5-25: Spatial level: a) Lower floor; b) Middle floor; c) Upper floor.

The third level of notation, code-named here as ‘diagrammatic level’, foregrounds

underlying, emergent boundaries of space and discards all connections between them. This level

of notation, closely related to the parti of a design, the geometrical diagram or pattern that

emerges when all details have been dropped out, is the most abstract version of the model and

functions as a scaffolding of the design. Metric distances between boundaries are taken into

account. The diagrammatic representation for all three floors is shown below in Figure 5-26.

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Figure 5-26: Diagrammatic level: a) Lower floor; b) Middle floor; c) Upper floor.

A closer look at the parsing of the model shows the application of a set of shape rules

that replace parts of the model with corresponding parts in the level below them. Some shape

rules are quite straightforward and they apply on simple continuous rectangular prisms with one

or more undulations on their boundaries or openings within them to denote openings. In most of

these cases the topology of the rectangular prisms is genus-0 (no interior holes) or genus-1 (one

interior hole), typically associated with a window for a vertical rectangular prism denoting a wall

or a staircase for a horizontal rectangular prism to denote a floor plate. A sample of these simple

rules for the three levels of representations of this model is shown in Figure 5-27.

Figure 5-27: Successive abstraction of simple wall elements

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The simple rules shown above can be combined with one another to describe more

complex spaces bounded by rectangular prisms. Often the interpretations and the parsing of the

design are considerably harder than that and the straightforward application of simple boundary

conditions cannot capture some of the subtleties of the design. In these cases the rules are more

complex too and refer primarily to dihedral space conditions where the faces of the three-

dimensional shapes turn to bound convex space. These latter cases involve decompositions of

complex architectural arrangements in sets of maximum numbers of large convex spaces (Peponis

and al 1997). A sample of these complex rules for the three levels of representations of this model

is shown in Figure 5-28.

Figure 5-28: Successive abstraction of complex wall elements: Left column: Architectonics level;

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The realignment of the plans of the house in terms of successive levels of abstraction

suggests an alternative reading for each floor that open up the issue of complexity to simplicity in

terms of subtracted information from the drawings. The previous reading of the house in terms of

parallel representations and computations, that is, all levels of the house given in a singular

manner, say, architectonic, spatial or diagrammatic, aspired to a coherent totality of a

representational mode. The realignment of these representations in terms of spatial indexing

privileges now a comparative contextualization of the house. Furthermore this relationship

establishes straightforward numerical relationships between the numbers of objects modeled in

the corresponding three-dimensional models of the house. For example, the number of objects

depicted for the first floor of the Smith House is eighty, forty-nine, and twenty eight for the

architectonic, spatial and diagrammatic level respectively and are shown in Figure 5-29.

Figure 5-29: Smith House first floor - notations: a) Architectonic; b) Spatial; c) Diagrammatic.

The number of objects depicted for the second floor of the Smith House is seventy

eight, fifty, and thirty four for the architectonic, spatial and diagrammatic level respectively and

are shown in Figure 5-30.

Figure 5-30: Smith House second floor - notations: a) Architectonic; b) Spatial; c) Diagrammatic.

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The number of objects depicted for the third floor of the Smith House is seventy two,

thirty six, and twenty eight for the architectonic, spatial and diagrammatic level respectively and

are shown in Figure 5-31.

Figure 5-31: Smith House third floor - notations: a) Architectonic; b) Spatial; c) Diagrammatic.

Lastly, the number of objects depicted for the first floor of the Smith House is twenty,

seventeen and seventeen for the architectonic, spatial and diagrammatic level respectively and are

shown in Figure 5-32.

Figure 5-32: Smith House Terrace - notations: a) Architectonic; b) Spatial; c) Diagrammatic.

5.5. Play: Partitions

‘Alongside the visual symmetry, we should imagine another one, out of which the

former only ‘shines forth.’ Plotinus in (Panofsky 1968)

The basic mechanism to abstract the elements of the house and foreground their relationships as

they are translated from level to level has been put in place. What is interesting in this process is

the re-working of the compositional machinery of the design and the exploration of the

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possibilities that this system allows with respect to the symmetry parts. This section starts with

the extraction of the five subgroups of the rectangle in order to identify the symmetric

transformations needed for this description. Additional marks concerning shape, weight, and

projection are added to yield the organizational scaffolding of the house.

These diagrams provide the underlying structures for the partition of the Smith House

into five classes, each corresponding to a unique subgroup of the structure of the rectangle. Each

of these five partitions of the geometry can occur at any floor of the house, -first, second or third,

and for any level of representation, architectonic, spatial and diagrammatic. The total number of

partitioning n then for this case study is forty five distinct ones. Still, what is attempted here uses

these methodological tools but in a new context. The key idea here is that the decomposition of

the geometry of the house occurs only at the diagrammatic level and this partitioning will specify

how the higher level notations would correspond to that. The total number of drawings is still the

same as before, forty-five, but there is a major qualitative shift in the analysis of information. A

little application of a symmetry analysis on the architectonics notation will produce descriptions

primarily characterized by asymmetrical information. The transformed application of the

symmetry analysis here of the architectonic level will pick up elements and conditions that would

have gone unnoticed before. The complete visual computation of all forty-five symmetry

partitioning for the Smith house for each floor is given in the Figure 5-33 through Figure 5-37.

All notations are read in sets of nine and are read from the lower right side to the upper left. Each

row is read from right to left and each column from bottom to up. The lower row represents the

analysis of the first floor, the middle row that of the second floor and the top row the analysis of

the third floor. The right column shows the diagrammatic notation, the middle column the spatial

and the left column the architectonic.

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Figure 5-33 (d): Dihedral (D2) symmetries of the Smith House at the diagrammatic level and their correspondences at the spatial and architectonic levels. a) 3rd floor; b) 2nd floor; c) 1st floor

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Figure 5-34 (a): Vertical (V) symmetries of the Smith House with | without D2

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Figure 5-35 (b): Horizontal (H) symmetries of the Smith House

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Figure 5-36 (c): Rotational (S) symmetries of the Smith House

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Figure 5-37 (e): Two Views of Unlike Elements of the Smith House at the diagrammatic level and

their correspondences at the spatial and architectonic levels. a) 3rd floor; b) 2nd floor; c) 1st floor

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5.6. Forward: Ordering

By extracting sub-shapes which maximize the representation of a particular sub-symmetry or a

combination of some of them, it is now possible to construct a partial order lattice or semi-lattice

to illustrate the overlay of symmetries involved. With these nested underlying structures for the

description of the design, a perceptual interface becomes accessible through non-visual functional

relations underlying the visual features appear. The following diagram illustrates all possible

subgroups of symmetry: three of them with two elements, and one, the identity, with one element.

The structure of the diagram can be accounted for in two ways: from top to bottom, sub-

symmetries are subtracted from the full symmetry of the rectangle; and conversely, from the

bottom to top, sub-symmetries are added to achieve higher orders of symmetry. Such a reading is

analogous to a lattice diagram of a partially ordered set, or sub-shapes of a shape.

The partial order lattice may offer the essential representation of the structure of the

Smith House complex spatial configuration. The representation of this structure by means of

modeling requires the determination of levels of abstraction where the operations take place

according to the logic of the design. If the lattice reveals the deep structure of the object, we still

need to refine the whole semantics that is carried out throughout the representation. Thus, the

definition of the levels of abstraction requires that the symmetry operations carry a kind of

representation that embeds some semantics. Thus, with three different levels of abstraction, we

can generate three levels of detailed lattices. As example, the Figure 5-38 through Figure 5-40

shows the implementation of an arbitrary lattice: example of the level 2 abstraction. The spatial

notation is omitted for clarity of representation.

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Figure 5-38 (a): Partial order lattice of the sub-symmetries of the first floor of the Smith House.

124

Figure 5-39(b): Partial order lattice of the sub-symmetries of the second floor of the Smith House.

125

Figure 5-40 (c): Partial order lattice of the sub-symmetries of the third floor of the Smith House.

126

5.7. Fast Forward: The Smith House recombinant

The sub-symmetry analysis has shown all the possible symmetrical correspondences that can be

drawn in the Smith house. Furthermore these relationships were ordered in partial ordered lattices

to show how these relationships are nested in specific ways one within the other. Here a

somewhat different approach is taken and its major focus is the juxtaposition of all these

correspondences, one with the other, to examine partial group theoretic descriptions of the Smith

house and in the way of doing so foreground specific relationships that a straightforward

application of group theory wouldn’t do.

The lattice of the rectangle consists of five subgroups. These subgroups can be

combined one with another to comprise a set of 25 = 32 possible design worlds that are

augmented each three drawings corresponding to the three floors of the house, making then a total

of ninety six drawings including the empty set. This recombinant vision exhausts all possible

ways that the parts of the house can be combined and should therefore be able to capture all

theoretical statements on symmetry that have been said or could be said abut the house. One of

the most important principles of modernism –and neo-modernism is its overt negation of the

obvious; symmetries are to be avoided if the only thing they do is that bring attention to

themselves. Still, it is argued here that this partial, incomplete and ambiguous rework of classical

principles of compositions such as symmetry, can indeed be discussed inter ms of the very same

tools that describe the straightforward applications of these tools. Here then it is suggested that

the partial and incomplete correspondences that may be observed in the Smith House should be

captured in any of the ninety-six subsets of the complete set of transformations of the smith

house. The complete list of the diagrams is given in Figure 5-41 and the complete set of all

drawings for all floors and any recombination is given in Figure 5-42 through Figure 5-47.

Figure 5-41: A complete list of all combinations of symmetry parts of the Smith house

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Figure 5-42 (a): A complete list of all combinations of symmetry parts – top down: DSHC | VSHC | DVSHC | DVSC | DVHC

128

Figure 5-43 (b): A complete list of all combinations of symmetry parts – top down: DVSH | DVH | DSH | DVS | SHC

129

Figure 5-44 (c): A complete list of all combinations of symmetry parts – top down: VHC | VSC | DHC | DVC | DSC

130

Figure 5-45 (d): A complete list of all combinations of symmetry parts – top down: VC | SC | HC | DS | DH

131

Figure 5-46 (e): A complete list of all combinations of symmetry parts – top down: DV | S | H | V | DC

132

Figure 5-47 (f): A complete list of all combinations of symmetry parts – top down: D | C

5.8. Discussion

An application of the methodology of formal analysis was attempted here using Richard Meier’s

Smith House as a case study. The history and some competing analytical approaches were

presented in the first part of the chapter and the second took the preliminary findings of the

analysis to produce a formal model of the house. Three levels of notations were used to tackle

constructs of composition such as layering, transparency, and collage. All plans of the house were

represented and decomposed in specific ways as described in the previous chapter and the

computation of all symmetry parts took place in entirely visual terms. A final reassembly of the

layered symmetries explained the structure of the symmetry of the house and provides an

illustration of one of the basic arguments of this thesis on the foundation of a theory of emergence

based on symmetry considerations.

A major challenge for the evaluation of the analysis is the degree that the

decompositions provided proved visually the established discourses on the house. The degree to

which these representations corroborate existing discourses on the Smith House provides a more

contested territory and this is an aspect of further critical research on the ability of the proposed

methodology to align itself with existing analytical discourses and prove them or not. Here the

implementation on Meier’s Smith House has nicely demonstrated a simple but quite significant

fact, the possibility of using formal tools from group theory and lattice theory to discuss

symmetry properties of designs that do not yield immediately their underlying structure.

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Content

Chapter 5 Visual computations..................................................................88

5.1. Introduction....................................................................................................... 88

5.2. White geometries .............................................................................................. 90

5.3. The Smith House............................................................................................... 94

5.3.1. A first encounter: Site Structuring ........................................................................... 95 5.3.2. Second encounter: Maximal lines ............................................................................ 97 5.3.3. Third encounter: Planes and walls ......................................................................... 101 5.3.4. Fourth encounter: Layers ....................................................................................... 107

5.4. The Smith House: A formal description ......................................................... 109

5.4.1. Initial shape ............................................................................................................ 110 5.4.2. Rewind ................................................................................................................... 110

5.5. Play: Partitions ................................................................................................ 115

5.6. Forward: Ordering .......................................................................................... 122

5.7. Fast Forward: The Smith House recombinant ................................................ 126

5.8. Discussion....................................................................................................... 132